Would a mathematical proof (i.e. not simply thought experiments implying moving magnets) that the Maxwell equations are not covariant be that
"The Maxwell equation in "potential form" under the Lorentz gauge of the scalar potential is...
[snip]

First, to do this correctly, one has to really spell out what is meant by the "Maxwell Equations", as well as specify the field variables and their constitutive equations.

You may enjoy these:
"If Maxwell had worked between Ampere and Faraday:
An historical fable with a pedagogical moral"
Max Jammer and John Stachel
Am. J. Phy, v48, no 1, Jan 1980, pp 5-7

I don't see what you mean. It failed under the Lorentz gauge, so that's all one need. No need to show it fails under other gauges. What would you think it takes to do a good demonstration?

As I said, I think that the problem has to be spelled out better. For example, there are "Maxwell equations" with E,B and equations with E,D,B,H. Using a potential formulation, you additionally deal with gauge issues. In addition, the way these fields transform have to be specified. (It may be that these "ground rules" are implicit in your problem.)

The references above point out that there is a way that the Galilean transformations can be compatible with a subset of electromagnetic phenomena, which has some pedagogical value. (See the Jammer/Stachel paper: http://scitation.aip.org/getabs/ser...00048000001000005000001&idtype=cvips&gifs=Yes . Here is the abstract: If one drops the Faraday induction term from Maxwell's equations, they become exactly Galilei invariant. This suggests that if Maxwell had worked between Ampère and Faraday, he could have developed this Galilei-invariant electromagnetic theory so that Faraday's discovery would have confronted physicists with the dilemma: give up the Galileian relativity principle for electromagnetism (ether hypothesis), or modify it (special relativity). This suggests a new pedagogical approach to electromagnetic theory, in which the displacement current and the Galileian relativity principle are introduced before the induction term is discussed. This reference may suggest one way to address your question.)

Note also the title of the van Dantzig paper, which formulates Maxwell equations using differential forms without the use of a metric, implying compatibility with either Galilean or Lorentz transformations.